The invention relates in general to the field of computerized methods for performing blind calibration of sensor arrays, where signals from such arrays are subject to beamforming, in particular in the fields of radio interferometry (to recover a sky image), magnetic resonance imaging or ultrasound imaging.
Image reconstruction from signals received by sensor arrays is used in many application fields, including radio interferometry for astronomical investigations, and magnetic resonance imaging, ultrasound imaging, and positron emission tomography for medical applications.
For example, modern large-scale radio telescope arrays use antenna stations composed of multiple antennas that are closely placed for imaging the sky. The signals received by the antennas at a station are combined by beamforming to reduce the amount of data to be processed in the later stages. Currently, beamforming at antenna stations is typically done by conjugate matched beamforming towards the center of the field of view at all antenna stations. The signals transmitted by the stations are then correlated to obtain measurement values called visibilities, which roughly correspond to the samples of the Fourier transform of the sky image. The reconstruction of the sky image is thus obtained from methods based on the inverse Fourier transform of the entire collection of visibility measurements.
The instruments for the above applications often have a hierarchical system architecture in the sense that they are phased-arrays of several smaller phased-arrays (groups of compact sensor elements acting as receivers) called stations (or subarrays). Consequently, beamforming techniques adopted within these instruments may also follow the same hierarchy, performed initially for individual stations, and later on for the whole instrument. A suitable station calibration prior to beamforming allows for better exploitation of the instrument. This calibration estimates amplitude adjustment and phase shift compensation parameters for each individual sensor gain, correcting for system losses and delays in station measurements. The most popular methods are of the supervised variety: they rely on known properties of known sources to estimate instrumental unknown parameters. However, such methods have two main drawbacks: (a) prior information about the region of interest is available only for strong sources, which leads to loss in performance because weak sources are disregarded, and (b) performance is sensitive to the accuracy of strong source data.
In addition, blind calibration methods exist, that is, unsupervised methods, where calibration works without resorting to known sources, such that the above difficulties are circumvented. Two types of blind calibration approaches are known. The first approach, called redundancy calibration, makes explicit use of redundant baselines (those baselines having same length and orientation), to repeatedly observe the same resultant Fourier sample of the region of interest. With sufficient groups of redundant baselines, the sensor element gains can be estimated more accurately and faster than with supervised calibration methods. Such a scheme, however, requires deploying sensor elements that are entirely devoted to this task, rather than using them for further baselines. The second blind calibration approach uses convex optimization or message passing algorithms and relies on the fact that, at the low signal to noise ratio (SNR) levels that are typically found in station observations, there are only a few strong sources detectable, with weaker ones buried in the noise. Consequently, the sources can be assumed to be sparse and blind station calibration can be formulated as a general sparsity problem, which aims to estimate signals along with associated instrumental amplitude and phase distortions.